We revisit two well-studied problems, Bounded Degree Vertex Deletion and Defective Coloring, where the input is a graph $G$ and a target degree $\Delta$ and we are asked either to edit or partition the graph so that the maximum degree becomes bounded by $\Delta$. Both are known to be parameterized intractable for treewidth. We revisit the parameterization by treewidth, as well as several related parameters and present a more fine-grained picture of the complexity of both problems. Both admit straightforward DP algorithms with table sizes $(\Delta+2)^\mathrm{tw}$ and $(\chi_\mathrm{d}(\Delta+1))^{\mathrm{tw}}$ respectively, where tw is the input graph's treewidth and $\chi_\mathrm{d}$ the number of available colors. We show that both algorithms are optimal under SETH, even if we replace treewidth by pathwidth. Along the way, we also obtain an algorithm for Defective Coloring with complexity quasi-linear in the table size, thus settling the complexity of both problems for these parameters. We then consider the more restricted parameter tree-depth, and bridge the gap left by known lower bounds, by showing that neither problem can be solved in time $n^{o(\mathrm{td})}$ under ETH. In order to do so, we employ a recursive low tree-depth construction that may be of independent interest. Finally, we show that for both problems, an $\mathrm{vc}^{o(\mathrm{vc})}$ algorithm would violate ETH, thus already known algorithms are optimal. Our proof relies on a new application of the technique of $d$-detecting families introduced by Bonamy et al. Our results, although mostly negative in nature, paint a clear picture regarding the complexity of both problems in the landscape of parameterized complexity, since in all cases we provide essentially matching upper and lower bounds.

翻译：我们重新研究了两个经典问题——有界度顶点删除和缺陷染色，其中输入图$G$和目标度数$\Delta$要求通过编辑或划分操作使得图的最大度不超过$\Delta$。已知这两个问题对于树宽是参数化不可解的。我们重新审视了树宽参数化及其相关参数，并呈现了这两个问题复杂度的更精细化图景。两个问题均存在直接的动态规划算法，其表大小分别为$(\Delta+2)^{\mathrm{tw}}$和$(\chi_\mathrm{d}(\Delta+1))^{\mathrm{tw}}$，其中tw是输入图的树宽，$\chi_\mathrm{d}$为可用颜色数。我们证明在SETH假设下，即使将树宽替换为路径宽，这两个算法都是最优的。在此过程中，我们还获得了一个缺陷染色算法，其复杂度在表大小上呈拟线性，从而确定了这两个问题在这些参数下的复杂度。随后我们考虑更受限的参数树深度，并填补已知下界之间的空白，证明在ETH假设下，两个问题均无法在$n^{o(\mathrm{td})}$时间内求解。为此，我们采用了一种可能具有独立价值的递归低树深度构造方法。最后，我们证明对这两个问题，$\mathrm{vc}^{o(\mathrm{vc})}$算法将违反ETH假设，因此已有算法已是最优的。证明依赖于Bonamy等人引入的$d$-检测族技术的新应用。尽管我们的结果本质上是负面的，但它们清晰描绘了参数化复杂度领域中这两个问题的全貌——在所有情况下我们均提供了本质上匹配的上界和下界。